One formula calculates the longitudinal angle between the observer and the Sun (subpoint); you would combine that with your longitude and look in an ephemeris or navigational almanac for the time when the Sun is predicted to have that GHA. The formula is an algebraic transformation of this equation: Sin height = (sin lat sin dec) + (cos lat cos dec cos LHA). height = angular height (from horizontal) lat = latitude dec = declination LHA = Local Hour Angle, the difference in the longitudinal coordinates of the observer and the celestial body, e.g. Sun. Cos LHA = [(sin height) - (sin lat sin dec)]/(cos lat cos dec). The height of the Sun at sunrise is estimated. It will be about the sum of its semidiameter (SD) correction (SD is within .3' of its -16' average), nominal astronomical refraction (about -34'), and the ardistance [-1.15'root(elevation of observer in feet)] between the horizon and the observer. {That arcdistance is the apparent dip [about -.97'root(elevation of observer in feet)] + the terrestrial refraction correction of (.97' - 1.15')root(elevation of observer)}. So the sine of the height is determined and can be entered as a constant in the formula. For 10,000 feet, the height would be about -(16' + 34' + 115') or -165' or -(2 deg 45 min). Its sine is about -.047978. For example: LHA = arccos [-.047978 - (sin lat sin dec)]/(cos lat cos dec). When evaluated it tells the difference in longitude between the observer and the Sun's subpoint on Earth. If, for example the difference is 95 deg 02 min, and the observer is at 120 deg 30 min West, then the Sun's Greenwich Hour Angle (GHA) is (120 deg 30 min) - (95 deg 02 min) = 025 deg 28 min. Since the GHA of the Sun is tabulated (in the Online Nautical Almanac, for instance), the (sunrise) moment can be found (by interpolating) when the Sun is predicted to have a GHA of 025 deg 28 min. Mark Prange
